Density-functional theory

Density-functional theory is a set of theories in statistical mechanics that profit from the
fact that the Helmholtz energy function of a system can be cast as a functional of
the density. That is, the density (in its usual sense of particles
per volume), which is a function of the position in inhomogeneous systems,
uniquely defines the Helmholtz energy. By minimizing this Helmholtz energy one
arrives at the true Helmholtz energy of the system and the equilibrium
density function. The situation
parallels the better known electronic density functional theory,
in which the energy of a quantum system is shown to be a functional
of the electronic density (see the theorems by Hohenberg, Kohn, Sham, and Mermin).

Starting from this fact, approximations are usually made in order
to approach the true functional of a given system. An important
division is made between local and weighed theories.
In a local density theory the
in which the dependence is local, as exemplified by the (exact)
Helmholtz energy of an ideal system:

where is an external potential. It is an easy exercise
to show that Boltzmann's barometric law follows from minimization.
An example of a weighed density theory would be the
(also exact) excess Helmholtz energy for a system
of 1-dimensional hard rods:

where ,
precisely an average of the density over the length of
the hard rods, . "Excess" means "over
ideal", i.e., it is the total
that is to be minimized.